The weighted boundedness considerably less is known compared with the unweighted one. As already mentioned in Remark REF  (3), Beceanu [1]} obtained some weighted \(L^p\) -boundedness with polynomial weights \({\langle }x{\rangle }^a\) . Note that Beceanu–Schlag [2]}, [3]} proved (again for the Schrödinger operator) \(W_\pm \in \mathbb {B}(X)\)  if \(X\)  is any Banach space of measurable functions on \({\mathbb {R}}^3\)  such that the norm \({\Vert \cdot \Vert }_X\)  is invariant under reflections and translations and that
\({\Vert \chi _H f\Vert }_X\le A {\Vert f\Vert }_{X}\)  for any half space \(H\subset {\mathbb {R}}^3\)  with some uniform constant \(A\) . This result clearly implies the \(L^p\) -boundedness (even the \(W^{s,p}\) -boundedness) of \(W_\pm \) , but not the weighted \(L^p\) -boundedness since weighted \(L^p\) -norms are not invariant under translations.
